Theorem ralsd 50450

Description: Introduction rule for "all some" restricted to a class. This is the converse of rals1d 50453 and rals2d 50454 taken together. (Contributed by David A. Wheeler, 12-Jul-2026.)

Hypotheses

Ref	Expression
ralsd.1	⊢ (𝜑 → ∀𝑥 ∈ 𝐴 (𝜓 → 𝜒))
ralsd.2	⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝜓)

Assertion

Ref	Expression
ralsd	⊢ (𝜑 → ∀∃𝑥 ∈ 𝐴(𝜓 → 𝜒))

Proof of Theorem ralsd

Step	Hyp	Ref	Expression
1		ralsd.1	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 (𝜓 → 𝜒))
2		ralsd.2	. 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝜓)
3		df-rals 50447	. 2 ⊢ (∀∃𝑥 ∈ 𝐴(𝜓 → 𝜒) ↔ (∀𝑥 ∈ 𝐴 (𝜓 → 𝜒) ∧ ∃𝑥 ∈ 𝐴 𝜓))
4	1, 2, 3	sylanbrc 594	1 ⊢ (𝜑 → ∀∃𝑥 ∈ 𝐴(𝜓 → 𝜒))

Syntax hints:   → wi 4  ∀wral 3085  ∃wrex 3095  ∀∃wrals 50445

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 210  df-an 401  df-rals 50447

This theorem is referenced by: (None)